What if I told you that everywhere you point--as well as at every spot you can imagine, be it the tip of your pen, the tip of your index finger, or the outermost tip of Cape Horn-- there is a hidden, six-dimensional space so small that we cannot see it nor ever set foot in it? Yet it's there, all the same, and pretty much rules our universe. Perhaps you'd say I was crazy, especially since I've spent the last four years learning everything I could about this strange realm and even coauthored a book about it, *The Shape of Inner Space*, even though the space in question may not be there at all.

Before you lock me away, I'd like to point out that there are a number of other equally crazy-sounding ideas about the universe that are taken quite seriously. Such as the notion, popular in quantum cosmology circles, that the universe started from nothing. Or that it started from nothing, was converted to something, and then changed into something else all over again. And it all will change again (and not for the better) before we're done here. Then there's the apparent fact that, after years and centuries of study, we have no clue as to what 96 percent of the universe consists of. Although we have a catchy name for this stuff--dark energy and dark matter--we don't know what it is.

As for that business of the "extra" dimensions, which is central to this book, I can say, in my defense, that I didn't make up the idea. It's a basic premise of string theory (and its close relative, M-theory)--regarded by many observers as the most promising attempt yet to understand how the universe works and account for all the particles and forces of nature. Sometimes called the theory of everything, string theory is supposed to provide an all-encompassing blueprint not only for our universe but for a limitless number of other universes that might possibly exist alongside ours.

That's why I've spent so much time ensconced in the six-dimensional realm of "Calabi-Yau" space, as this geometric construct is called. Theorists believe that the shape, or geometry, of this tiny, curled-up domain dictates the kind of universe we live in--the kind of particles that exist and the interactions between them. If mathematicians and physicists can figure out the exact shape of these additional six dimensions--the "shape of inner space," so to speak--they may unlock the key to our universe. Different shapes, according to this school of thought, correspond to different universes endowed with different physical properties.

No good argument has been advanced so far as to why these other universes cannot exist, though there is, of course, absolutely no proof that they do. That holds for string theory as well. There is not a shred of evidence to support the contention that all particles, at their most basic level, are made up of submicroscopic strings, trillions of times smaller than an electron. Nor can anyone back up the proposition that these same strings, out of which our universe is presumably built, move and vibrate in a 10-dimensional space--and perhaps in an 11-dimensional space too. It's called string theory for a reason, because it is just a "theory," albeit a very ambitious one. (And in some circles a very dubious one.)

The physics side of this picture is open to debate, and partisans on both sides have been going at it for quite some time. But there's also the mathematics side of things, where once something is proven--assuming it's done right--there's little room for discussion. That verdict will stand forever, and it is not subject to change based on the outcome of future experiments. My entrée to this subject came from the mathematics end through my association with Harvard mathematician Shing-Tung Yau, my coauthor on this book. When Yau started working on Calabi-Yau spaces (as they were dubbed years later), he'd never heard of string theory before, and the field, in its present incarnation, did not exist. Instead, he was intrigued by a conjecture posed by the geometer Eugenio Calabi, who spoke of complex, higher-dimensional spaces with very special properties. Physics did not enter into it so far as Calabi was concerned.

Yau proved that these complicated spaces, or shapes, exist *mathematically*. While that doesn't tell us how things work in the "real world," he had a strong sense that this geometry would be important for physics. He felt that partly because the equations for the Calabi-Yau shapes were related to Einstein's equations of motion and gravity. But Yau was also guided by the conviction that truly deep results in mathematics must have some manifestation in nature. Understanding the universe, he's always maintained, is not just the province of physicists. Mathematicians contribute to this enterprise too, and he frequently urged physics colleagues to see where his geometric forms might fit in.

But Yau had to wait eight years for physicists to discover his theorem and put the shapes that he proved possible at the center of string theory. Since that time, about a quarter of a century ago, an exhilarating collaboration between mathematicians and physicists has unfolded. New advances in mathematics have spurred new ideas in physics, which have led, in turn, to new results in mathematics, some of which have been quite spectacular. While string theory remains not only unproven but empirically untested, it has passed the test of mathematical consistency. That's an essential first step, because if the numbers did not add up, we would have tossed out string theory long ago. But it's gone much farther than that, as string theory is now tied into diverse areas of mathematics, which are solid and proven beyond doubt, thereby helping to

forge intriguing new connections between previously disparate mathematical fields. These connections, inspired by string theory, have been so rich and fruitful as to make us think that the physicists must be onto something after all. Which means we have to consider the possibility that among the crazy ideas put forth to explain the universe, this one might be crazy enough to be true.